Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators

Kaori Nagatou, Michael Plum, Mitsuhiro T. Nakao

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Subject of investigation in this paper is a one-dimensional Schrödinger equation, where the potential is a sum of a periodic function and a perturbation decaying at ±∞. It is well known that the essential spectrum consists of spectral bands, and that there may or may not be additional eigenvalues below the lowest band or in the gaps between the bands. While enclosures for gap eigenvalues can comparatively easily be obtained from numerical approximations, e.g. by D. Weinstein's bounds, there seems to be no method available so far which is able to exclude eigenvalues in spectral gaps, i.e. which identifies subregions (of a gap) which contain no eigenvalues. Here, we propose such a method. It makes heavy use of computer assistance; nevertheless, the results are completely rigorous in the strict mathematical sense, because all computational errors are taken into account.

Original languageEnglish
Pages (from-to)545-562
Number of pages18
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume468
Issue number2138
DOIs
Publication statusPublished - 2012 Feb 8
Externally publishedYes

Fingerprint

Enclosures
eigenvalues
Eigenvalue
operators
Operator
periodic functions
Spectral Gap
Essential Spectrum
Enclosure
spectral bands
enclosure
Periodic Functions
Numerical Approximation
Lowest
Perturbation
perturbation
approximation

Keywords

  • Eigenvalue excluding
  • Error estimates
  • Schrödinger operator

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

Cite this

Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators. / Nagatou, Kaori; Plum, Michael; Nakao, Mitsuhiro T.

In: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 468, No. 2138, 08.02.2012, p. 545-562.

Research output: Contribution to journalArticle

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